JEE Mathematics 2023 Question with Solution

Given: The parabola is $y^2 = 36x$, so in standard form $y^2 = 4ax$ we have $a = 9$. Also, $PQ$ is a focal chord of length $100$, the ordinate of $P$ is positive, and $M$ divides $PQ$ in the ratio $3:1$.

Find: Which given point does not lie on the line through $M$ perpendicular to $PQ$.

For the parabola $y^2 = 4ax$, a parametric point is $\left(at^2, 2at\right)$. Hence here

and since $PQ$ is a focal chord, the other point is obtained using $t_1 t_2 = -1$, so

Using the given length condition $PQ = 100$ and the fact that the ordinate of $P$ is positive, we get

Now $M$ divides $PQ$ internally in the ratio $PM:MQ = 3:1$. Therefore,

Substituting $P = (81, 54)$ and $Q = (1, -6)$,

The slope of $PQ$ is

So the slope of the perpendicular line is

Through $M = (21, 9)$, its equation is

which simplifies to

Now check the options:

• For $(3, 33)$: $y = -\frac{4}{3}(3) + 37 = 33$, so it lies on the line.
• For $(6, 29)$: $y = -\frac{4}{3}(6) + 37 = 29$, so it lies on the line.
• For $(-6, 45)$: $y = -\frac{4}{3}(-6) + 37 = 45$, so it lies on the line.
• For $(-3, 43)$: $y = -\frac{4}{3}(-3) + 37 = 41 \ne 43$, so it does not lie on the line.

Therefore, the point that does not lie on the required line is $(-3, 43)$.

The solution states The Correct Option is C, but the actual point found from the working is $(-3, 43)$, which corresponds to option D in the listed options. Since the options do not match the solution label, the defensible answer from the working is D.